Why are $T^k(V)$ and $V^* \otimes... \otimes V^*$ just isomorphic?

Question

In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* \otimes... \otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $\epsilon^1,...,\epsilon^n$ is its dual basis than $\epsilon^{i_1}\otimes...\otimes\epsilon^{i_n}$ is a basis of both $T^k(V)$ and $V^* \otimes... \otimes V^*$ no? So why just an isomorphism?